relations in symplectomorphism groups and degenerations of lg

Relations in symplectomorphism groups anddegenerations of LG models

Gabriel Kerr

joint with C. Diemer and L. Katzarkov

May 3, 2012

Gabriel Kerr (University of Miami) Relations in symplectomorphism groups and degenerations of LG modelsMay 3, 2012 1 / 38

Outline

1 Introduction and motivation

2 Review of secondary stacks

3 Generators of GAToric hypersurface degenerationsStratified Morse functions

4 Relations in GADefinition and basic properties of circuitsExamples in dimension 1Vanishing cycle in circuit hypersurfaces

5 ApplicationsLandau-Ginzburg ModuliExampleHMS of the Mori program

Outline

Mapping class group Map(Σ, ∂Σ) = π0(Diff or (Σ, ∂Σ))

Finitely generated by Dehn twists.

Relations given by examining basic model surfaces.

Several finite presentations exist.

Relations for Map(Σ, ∂Σ)

TT T T= TT

The braid relation

TT T = T∂Σ

The lantern relation

TT =T∂Σ)3(

The star relation

For H a symplectic orbifold with normal crossing divisor ∂H.

Overarching Goal:

Describe decompositions of H.

Examine various generators of π0(Symp(H, ∂H)) via symplecticmonodromy.

Find relations occurring in fundamental decomposed pieces of H.

Apply results to group presentations and LG models.

Overarching Goal:

Outline

Let A ⊂ Zd be a finite set and Q = Conv(A) ⊂ Rd its convex hull.We define XA to be the toric stack defined by A and OA(1) itsequivariant line bundle.

X∨Σ

(A, Q)

Taking ω to be the symplectic form of XA with [ω] = c1(OA(1)), themoment map then is a map µA : XA → Q.

Let A ⊂ Zd be a finite set and Q = Conv(A) ⊂ Rd its convex hull.We define XA to be the toric stack defined by A and OA(1) itsequivariant line bundle.

X∨Σ

(A, Q)

Taking ω to be the symplectic form of XA with [ω] = c1(OA(1)), themoment map then is a map µA : XA → Q.

A smooth hypersurface H ⊂ XA defined by a sections ∈ H0(XA,OA(1)) has an amoeba equal to the image µA(H).

A subdivision of A defines a degeneration of (H,XA,OA(1)) whichsimultaneously subdivides the amoeba.

A maximal subdivision is a triangulation which defines a pair of pantsdecomposition for H.

There exists a toric stack XSec(A), called the secondary stack, whoseorbits correspond to regular subdivisions of the marked polytope(Q,A).

Theorem

There exists a toric moduli stack XSec(A) for toric hypersurfaces H andtheir degenerations with toric universal hypersurface:

π : H ⊂ XLaf (A) → XSec(A)

The moment polytope of XSec(A) is the secondary polytope Sec(A) and themoment polytope of XLaf (A) is a Minkowski sum of Sec(A) with a simplex.

This implies any toric degeneration Y → C of (H,XA,OA(1)) can berealized as the pullback of π along a map j : C→ XSec(A).

The combinatorics of Sec(A) dictates the geometry of thedegenerations.

The principal A-determinant is a section of OSec(A)(1) on XSec(A). Itszero locus EA determines hypersurfaces that are singular or intersectthe boundary divisors non-transversely.

Theorem

Choosing a base point ∗ ∈ XSec(A) and a global symplectic form onXLaf (A) − π−1(EA) for which ∂H is horizontal relative to π,symplectic parallel transport defines a map:

P : Ω∗(XSec(A) − EA)→ Symp(H, ∂H)

Take GA ≤ Symp(H, ∂H) to be the image.

Theorem

Choosing a base point ∗ ∈ XSec(A) and a global symplectic form onXLaf (A) − π−1(EA) for which ∂H is horizontal relative to π,symplectic parallel transport defines a map:

P : Ω∗(XSec(A) − EA)→ Symp(H, ∂H)

Take GA ≤ Symp(H, ∂H) to be the image.

Outline

The components of EA of two essentially different types.

The first type of symplectomorphism is obtained from monodromy ofa toric hypersurface around a toric degeneration, or a component ofEA on the boundary of XSec(A).

More precisely, taking A ⊂ Zd and η ∈ ZA, we consider a torichypersurface degeneration of (H,XA) defined by η pulled back alonga map fη : C→ XSec(A).

The red portion gives the degenerated amoeba. The blue portion describesthe region of smoothing along the discontinuity of the distinct toricactions.

Monodromy is described as a discrete torus action on the interiors ofdegenerated components convolved along a small neighborhood of thecoisotropic vanishing cycle.

Outline

The second type of monodromy map considered is that of a stratifiedMorse function.

The local model is a (U, ∂U) ⊂ (Cd+1,∪m≤i≤dHi ) whereHi = zi = 0 is the i-th coordinate hyperplane and Cd+1 is thestandard symplectic form. The function is given by:

fm(z0, . . . , zd) =m∑i=0

d∑i=m+1

If m = d , this is a Morse singularity in the usual sense. Otherwise thisgives rise to a stratified Morse function in the sense of Goresky andMacpherson.

fm(z0, . . . , zd) =m∑i=0

d∑i=m+1

fm(z0, . . . , zd) =m∑i=0

d∑i=m+1

The symplectomorphism is a generalized braid, or a Dehn twist about avanishing cycle with boundary.

d = 1,m = 1 d = 2,m = 2d = 2,m = 1

V ≈ S1 ? ∆0 V ≈ S0 ? ∆1

For general d and m, the vanishing cycle is the join

V ≈ Sd−m ?∆m−1

The key motivation for the consideration of the abovesymplectomorphisms arises from the following theorem:

Theorem

Let δ : S1 → XSec(A) − EA be a small loop around a smooth point of EA.Then symplectic monodromy of π : (H, ∂H)→ XSec(A) around δ is eitherthat of a stratified Morse function or a toric hypersurface degeneration.

Corollary

The group GA is generated by toric hypersurface degeneration andstratified Morse monodromy.

The key motivation for the consideration of the abovesymplectomorphisms arises from the following theorem:

Theorem

Let δ : S1 → XSec(A) − EA be a small loop around a smooth point of EA.Then symplectic monodromy of π : (H, ∂H)→ XSec(A) around δ is eitherthat of a stratified Morse function or a toric hypersurface degeneration.

Corollary

The group GA is generated by toric hypersurface degeneration andstratified Morse monodromy.

Outline

For relations we seek fundamental nontrivial pieces of degenerations. Thenext level of complexity after a simplex is known as a circuit:

Definition

Let A ⊂ Zd be a finite set.

(i) A is a circuit if every proper subset is affinely independent.

(ii) A is an extended circuit if its R affine span is Rd and the lattice ofaffine relations has rank 1

Every extended circuit A = α0, . . . , αd+1 is uniquely determined by theprimitive generator a = (a0, . . . , ad+1) ∈ Zd+2 of the lattice of affinerelations for A.

Definition

By definition a satisfies:

d+1∑i=0

aiαi = 0

d+1∑i=0

ai = 0

The signature σA = (p, q; r) of A denotes the number of positive,negative and zero entries in a. We say that A is non-degenerate ifr = 0 and write σA = (p, q).

The signature determines Q = Conv(A) combinatorially. Anon-degenerate (p, q) circuit is polar dual to a (skew) product ofsimplices ∆p−1 ×∆q−1.

This implies that the boundary ∂H of H ⊂ XA can be expressed asthe union of pq components.

d+1∑i=0

aiαi = 0

d+1∑i=0

ai = 0

d+1∑i=0

aiαi = 0

d+1∑i=0

ai = 0

d+1∑i=0

aiαi = 0

d+1∑i=0

ai = 0

Examples:

σA : (1, 3)

a : (−3, 1, 1, 1)

(1, 2; 1)

(−2, 1, 1, 0)(−1,−1, 1, 1)

(2, 2)

(2, 2; 1)a : (−4, 1, 1, 1, 1)σA : (1, 4) (2, 3)

(−3,−3, 2, 2, 2) (−1,−1, 1, 1, 0)

Every extended circuit admits exactly two coherent triangulations sothat its secondary polytope is a line segment and its secondary varietyis P1.

The Lafforgue stack XLaf (A) of A is a weighted blow up of Pd+1 alongthe codimension 2 coordinate subspaces defined by:

Z =⋃

ai<0,aj>0

Xi = 0 = Xj

The secondary stack of A is a weighted P(d−, d+) whered± = gcdai : ±ai > 0.The universal bundle and hypersurface H are given by the pull backof O(1) and proper transform of[X0 : · · · : Xd+1] :

∑d+1i=0 Xi = 0 ⊂ Pd+1.

Z =⋃

ai<0,aj>0

Xi = 0 = Xj

∑d+1i=0 Xi = 0 ⊂ Pd+1.

Z =⋃

ai<0,aj>0

Xi = 0 = Xj

The secondary stack of A is a weighted P(d−, d+) whered± = gcdai : ±ai > 0.

The universal bundle and hypersurface H are given by the pull backof O(1) and proper transform of[X0 : · · · : Xd+1] :

∑d+1i=0 Xi = 0 ⊂ Pd+1.

Z =⋃

ai<0,aj>0

Xi = 0 = Xj

∑d+1i=0 Xi = 0 ⊂ Pd+1.

When restricted to Pd+1 − Z , the map π : XLaf (A) → XSec(A) forcircuits can be viewed as a pencil defined as:

π([X0 : · · · : Xd+1]) =

∏ai>0

:∏aj<0

)−aj

The critical values of πH := π|H depend on the signature of A. Ifp 6= 1 6= q, then they are crit(π) = [0 : 1], [1 : 1], [1 : 0], while ifp = 1, they are crit(π) = [1 : 1], [1 : 0].In a neighborhood over [1 : 1], πH is a stratified Morse function withdegeneracy r . Over the [0 : 1], [1 : 0] critical values, πH is a toricdegeneration of H.

π([X0 : · · · : Xd+1]) =

∏ai>0

:∏aj<0

)−ajThe critical values of πH := π|H depend on the signature of A. Ifp 6= 1 6= q, then they are crit(π) = [0 : 1], [1 : 1], [1 : 0], while ifp = 1, they are crit(π) = [1 : 1], [1 : 0].

In a neighborhood over [1 : 1], πH is a stratified Morse function withdegeneracy r . Over the [0 : 1], [1 : 0] critical values, πH is a toricdegeneration of H.

π([X0 : · · · : Xd+1]) =

∏ai>0

:∏aj<0

)−ajThe critical values of πH := π|H depend on the signature of A. Ifp 6= 1 6= q, then they are crit(π) = [0 : 1], [1 : 1], [1 : 0], while ifp = 1, they are crit(π) = [1 : 1], [1 : 0].In a neighborhood over [1 : 1], πH is a stratified Morse function withdegeneracy r . Over the [0 : 1], [1 : 0] critical values, πH is a toricdegeneration of H.

Circuit Relation

Choosing a base point ∗ ∈ P(d−, d+), we take loops δ0, δ1, δ∞ aroundthe critical values and let Ti = P(δi ) ∈ GA. We denote T∂H for atwist T (ti ,j) about the boundary divisor of H of order ti ,j = 2π

around the Xi = 0 = Xj boundary component.

Theorem

If A is a circuit with p 6= 1 6= q then the following relations occur inπ0(GA):

T0T1T∞ = T∂H

if p = 1 and ai > 0, then the relation:

(T1T∞)ai = T ai∂H

Circuit Relation

Theorem

T0T1T∞ = T∂H

Circuit Relation

Theorem

T0T1T∞ = T∂H

Outline

Example 1: A = (0, 0), (1, 0), (0, 1), (−1,−1), σA = (1, 3).

[0 : 1] [1 : 1] [0 : 1]

XSec(A)

XLaf (A)

π−1H (q)

(T1T∞)3 = T 3∂H

Star relation

Example 1: A = (0, 0), (1, 0), (0, 1), (−1,−1), σA = (1, 3).

[0 : 1] [1 : 1] [0 : 1]

XSec(A)

XLaf (A)

π−1H (q)

(T1T∞)3 = T 3∂H

Star relation

Example 2: A = (0, 0), (1, 0), (0, 1), (1, 1), σA = (2, 2).

[0 : 1] [1 : 1] [0 : 1]

XSec(A)

XLaf (A)

π−1H (q)

Lantern relation

T0T1T∞ = T∂H

Example 2: A = (0, 0), (1, 0), (0, 1), (1, 1), σA = (2, 2).

[0 : 1] [1 : 1] [0 : 1]

XSec(A)

XLaf (A)

π−1H (q)

πLantern relation

T0T1T∞ = T∂H

Outline

Following Viro, as interpreted by Gelfand, Kapranov and Zelevinsky,one may obtain the real locus of H near the toric degeneration pointsup to isotopy by considering part of a suitable cover of its tropicalamoeba.

Also noted was that a (p, q) circuit modification yielded a Morsesurgery.

In our setup, this occurs as the theorem:

Theorem

The restriction of πH to the real points of H over R>0 ⊂ XSec(A) yields areal stratified Morse function of signature (q − 1, p − 1; r).

We interpret this tropically below and sketch the general procedurefor specifying the vanishing cycle.

Theorem

Let H ⊂ X Laf (A) be the complement of the fibers of

π : XLaf (A) → XSec(A) over [0 : 1] and [1 : 0].

Then X Laf (A) ≈ (C∗)d+1 and there is a commutative diagram ofmoment maps

(C∗)d+1 µ−−−−→ Rd+1 ⊂ Rd+2yπ y<a, >

C∗ log |z|2−−−−→ RHere Rd+1 is the affine hyperplane (c0, . . . , cd+1) :

∑ci = 0

reflecting the homogeneous quotient of (C∗)d+2 by the diagonalaction.

Let H ⊂ X Laf (A) be the complement of the fibers of

π : XLaf (A) → XSec(A) over [0 : 1] and [1 : 0].

Then X Laf (A) ≈ (C∗)d+1 and there is a commutative diagram ofmoment maps

(C∗)d+1 µ−−−−→ Rd+1 ⊂ Rd+2yπ y<a, >

C∗ log |z|2−−−−→ RHere Rd+1 is the affine hyperplane (c0, . . . , cd+1) :

∑ci = 0

reflecting the homogeneous quotient of (C∗)d+2 by the diagonalaction.

Recall that H = X0 + · · ·+ Xd = 0 ⊂ (C∗)d+1 which is known asa higher dimensional pair of pants.

σA = (2, 2)σA = (1, 3)σA = (1, 2; 1)

The tropical hypersurface of H = π−1H (q) then can be interpreted as

an affine section of H.Varying the affine section along parallel planes yields the tropicalversion of πH.

σA = (2, 2)σA = (1, 3)σA = (1, 2; 1)

σA = (2, 2)

σA = (1, 3)σA = (1, 2; 1)

an affine section of H.

Varying the affine section along parallel planes yields the tropicalversion of πH.

σA = (2, 2)

σA = (1, 3)

σA = (1, 2; 1)

σA = (2, 2)σA = (1, 3)

σA = (1, 2; 1)

σA = (2, 2)σA = (1, 3)

σA = (1, 2; 1)

Vanishing thimbles and cycles for signature (1, d + 1) case.

Tropical amoeba

The vanishing thimble of πH over [1,∞) ∈ XSec(A)(R) is simply thereal locus H0(R). The vanishing cycle maps to the spheresurrounding the hole in the amoeba near the degeneration point.

Tropical amoeba

The vanishing thimble of πH over (0, 1] ∈ XSec(A)(R) admits acoordinate description. The vanishing cycle has a point as its tropicalamoeba.

Coamoeba

Switching to the coamoeba, we observe the vanishing cycle as theprequotiented immersed Lagrangian.

Description of vanishing cycle of a nondegenerate (p, q) signaturecircuit as a join of (p, 1) and (1, q) cases

Example of a (3, 2) circuit.

Recall that the real (2, 1) Morse singularity can be pictured as amovie:

The tropical version of the (3, 2) circuit πH : H0 → C∗

In the tropical amoeba, we see the vanishing thimbles of a (3, 1) and(1, 2) circuit occurring as the unstable and stable manifolds.

coamoeba amoeba

amoeba coamoeba

The coamoeba vanishing cycles of the (3, 1) and (1, 2) combine withthe amoeba cycles through a join construction, filling the compactpiece of the tropical hypersurface and yielding the vanishing cycle.

Outline

LG models often occur as orbits of one parameter subgroupsG ⊂ (C∗)A in P(H0(XA,OA(1))). Omitting sections that are not fullgives an open curve in XSec(A) whose compactification is a 1-cycleW ⊂ XSec(A).

These LG models can then be thought of generically as mapsφ : P1 → XSec(A) in a certain 1-cycle class [W ], i.e. elements of thestack of twisted stable maps K0(XSec(A), [W ]).

The coarse space KA([W ]) can be realized as a toric quotient ofXSec(A) by G . This quotient is the toric variety associated to a fiberpolytope known as the monotone path polytope ΣγW (Sec(A)).

The space KA([W ]) parameterizes LG models and theirdegenerations. The fixed points of the torus action play the role oflarge complex limits by analogy to such points in XSec(A) reflectingthe LCL hypersurfaces.

Digression on monotone path polytopes:

Given a polytope Q ⊂ Rn and a linear map γ : Rn → R withγ(Q) = I , one defines the monotone path polytope Σγ(Q) to be thefiber polytope of γ : Q → I .

Dimension k faces of Σγ(Q) correspond to piecewise linear sections ofγ : Q → I with linear pieces contained in dimension at most (k + 1)faces of Q.

Vertices of Σγ(Q) consist of edge paths connecting the maximumface of Q relative to γ to the minimum.

Any such edge path for ΣγW (Sec(A)) is a sequence of circuits. Weconsider the pullback of H along such a path to be a maximaldegeneration of the LG model.

Outline

Example: Blp(P1 × P1)mir .

A = Newt(w)

A′ =

The homological mirror of the blow up of P1 × P1 at a point is theLG model w : (C∗)2 → C given by:

w(x , y) = c1x + c2y + c3x−1 + c4y−1 + c4x−1y−1 + c5

where the coefficients depend on the symplectic form on Blp(P1×P1).

Taking (C∗)2 ⊂ XA as the dense torus orbit, we have thatW ′ = Span(w , x0) gives pencil in H0(XA,OA(1)) and a projective lineW ⊂ XSec(A).

For generic form, the potential has 5 critical values corresponding tothe 5 intersection points of W with EA in XSec(A).

A = Newt(w)

A′ =

A = Newt(w)

A′ =

The secondary polytope of A:

Every twisted stable map φ ∈ K0(XSec(A), [W ]) has moment imagewhich is a curve through the blue dot.

The secondary polytope of A:

Every twisted stable map φ ∈ K0(XSec(A), [W ]) has moment imagewhich is a curve through the blue dot.

The monotone path polytope is the moment polytope for KA.

Each torus fixed point of KA([W ]) corresponds to an edge path alongSec(A), starting at the tip and ending on the facet obtained from thecoarse subdivision associated to forgetting 0 ∈ A.

These represent maximally degenerated LG models which, over eachcomponent of the twisted stable curve, is a partial Lefschetz fibrationassociated to a circuit.

Outline

For every maximally degenerated LG model, we construct asemi-orthogonal decomposition of the Fukaya-Seidel category:

C = 4C1 + 3C2 + C3

F(π−1(C ), π) = 〈E1,E2,E3,E4,T1,T2,T3,E5〉

C = 4C1 + 3C2 + C3

F(π−1(C ), π) = 〈E1,E2,E3,E4,T1,T2,T3,E5〉

C = 4C1 + 3C2 + C3

F(π−1(C ), π) = 〈E1,E2,E3,E4,T1,T2,T3,E5〉

Signature (1, d + 1) circuits are easily seen to be equivariantquotients of homological mirrors to weighted projective space.

A subclass of (2, 2) circuits have been proven to be mirror toweighted blowups of toric surfaces.

The homological mirror of a non-degenerate circuit is anticipated tobe the semi-orthogonal subcategory associated to an orthogonal of aweighted flip.

For degenerate circuits, we observe semi-orthogonal decompositionscorresponding to projective bundles.

This results in a conjectural A-model mirror to the Mori program for toricvarieties:

Blp(P1 × P1)→ P1 × P1 → P1

Blp(P1 × P1)→ F1 ≈ P(O(−1)⊕O)→ P1

Blp(P1 × P1)→ F1 ≈ Blq(P2)→ P2 → ∗

Assume (XA,w) is homologically mirror to XmirA . To every maximal

degeneration ψ of W into r circuits, there is a minimal model sequence(MMS):

XmirA = Xr

fr99K Xr−1 99K · · ·

f199K X0

called the mirror MMS to ψ.

Theorem

Every equivariant MMS of XmirA is the mirror MMS to a unique maximal

degeneration.

maximal LG degenerations of W l

vertices of ΣγW (Sec(A))l

minimal model sequences of XmirA

XmirA = Xr

fr99K Xr−1 99K · · ·

f199K X0

Theorem

degeneration.

XmirA = Xr

fr99K Xr−1 99K · · ·

f199K X0

Theorem

degeneration.

To every MMS of XmirA :

XmirA = Xr

fr99K Xr−1 99K · · ·

f199K X0

Kawamata has given a semi-orthogonal decomposition of Db(XmirA ) into r

subcategories.

Conjecture

Given any maximal degeneration ψ of W . Let:

Fuk((C∗)d ,w) = < T1, . . . , Tr >Db(Xmir

A ) = < S1, . . . ,Sr >

be the semi-orthogonal decompositions associated to ψ and its mirrorMMS. Then there exists an equivalence of triangulated categories:

Φψ : Fuk((C∗)d ,w)→ Db(XmirA )

which restricts to equivalences Φψ : Ti → Si for all 1 ≤ i ≤ r .

To every MMS of XmirA :

XmirA = Xr

fr99K Xr−1 99K · · ·

f199K X0

Kawamata has given a semi-orthogonal decomposition of Db(XmirA ) into r

subcategories.

Conjecture

Given any maximal degeneration ψ of W . Let:

Fuk((C∗)d ,w) = < T1, . . . , Tr >Db(Xmir

A ) = < S1, . . . ,Sr >

be the semi-orthogonal decompositions associated to ψ and its mirrorMMS. Then there exists an equivalence of triangulated categories:

Φψ : Fuk((C∗)d ,w)→ Db(XmirA )

which restricts to equivalences Φψ : Ti → Si for all 1 ≤ i ≤ r .

relations in symplectomorphism groups and degenerations of lg

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